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Trends in logic

The emergence of fuzzy events in Hilbert space quantum theory

M. Dalla Chiara, Roberto Giuntini, Richard J. Greechie

Published Jan 1, 2004• 0 citations
Publisher
DOI
10.1007/978-94-017-0526-4_4

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Reasoning in Quantum Theory: Sharp and Unsharp Quantum Logics

Chiara Manfriani, Roberto Giuntini, Richard J. Greechie

List of Figures. List of Tables. Preface. Acknowledgements. I: Mathematical and Physical Background. Introduction. 1. The mathematical scenario of quantum theory and von Neumann's axiomatization. 1.1. Algebraic structures. 1.2. The geometry of quantum theory. 1.3. The axiomatization of orthodox QT. 1.4. The 'logic' of the quantum events. 1.5. The logico-algebraic approach to QT. 2. Abstract axiomatic foundations of sharp QT. 2.1. Mackey's minimal axiomatization of QT. 2.2. Events. 2.3. Event-state systems. 2.4. Event-state systems and preclusivity spaces. 3. Back to Hilbert space. 3.1. Events as closed subspaces. 3.2. Events as projections. 3.3. Hilbert event-state systems. 3.4. From abstract orthoposets of events to Hilbert lattices. 4. The emergence of fuzzy events in Hilbert space quantum theory. 4.1. The notion of effect. 4.2. Effect-Brouwer Zadeh posets. 4.3. MacNeille completions. 4.4. Unsharp preclusivity spaces. 5. Effect algebras and quantum MV algebras. 5.1. Effect algebras and Brouwer Zadeh effect algebras. 5.2. The Lukasiewwicz operations. 5.3. MV algebras and QMV algebras. 5.4. Quasi-linear QMV algebras and effect algebras. 6. Abstract axiomatic foundations of unsharp quantum theory. 6.1. A minimal axiomatization of unsharp QT. 6.2. The algebraic structure of abstract effects. 6.3. The sharply dominating principle. 6.4. Abstract unsharp preclusivity spaces. 6.5. Sharp and unsharp abstract quantum theory. 7. To what extent is quantum ambiguity ambiguous? 7.1. Algebraic notions of 'sharp'. 7.2. Probabilistic definitions of 'sharpness'. II: Quantum Logics as Logic. Introduction. 8. Sharp quantum logics. 8.1. Algebraic and Kripkean semantics for sharp quantum logics. 8.2. Algebraic and Kripkean realizations of Hilbert event-state systems. 8.3. The implication problem in quantum logic. 8.4. Five polynomial conditions. 8.5. The quantum logical conditional as a counterfactual conditional. 8.6. Implication-connectives. 9. Metalogical properties and anomalies of quantum logic. 9.1. The failure of the Lindenbaum properety. 9.2. A modal interpretation of sharp quantum logics. 10. An axiomatization of OL and OQL. 10.1. The calculi of OL and OQL. 10.2. The soundness and completeness theorems. 11. The metalogical intractability of orthomodularity. 11.1. Orthomodularity is not elementary. 11.2. The embeddability problem. 11.3. Hilbert quantum logic and the orthomodular law. 12. First-order quantum logics and quantum set theories. 12.1. First-order semantics. 12.2. Quantum set theories. 13. Partial classical logic, the Lindenbaum property and the hidden variable problem. 13.1. Partial classical logic. 13.2. Partial classical logic and the Lindenbaum property. 13.3. States on partial Boolean algebras. 13.4. The Lindenbaum property and the hidden variable problem. 14. Unsharp quantum logics. 14.1.

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Elsevier eBooks•Jan 1, 2007•4 citations
List of Figures. List of Tables. Preface. Acknowledgements. I: Mathematical and Physical Background. Introduction. 1. The mathematical scenario of quantum theory and von Neumann's axiomatization. 1.1. Algebraic structures. 1.2. The geometry of quantum theory. 1.3. The axiomatization of orthodox QT. 1.4. The 'logic' of the quantum events. 1.5. The logico-algebraic approach to QT. 2. Abstract axiomatic foundations of sharp QT. 2.1. Mackey's minimal axiomatization of QT. 2.2. Events. 2.3. Event-state systems. 2.4. Event-state systems and preclusivity spaces. 3. Back to Hilbert space. 3.1. Events as closed subspaces. 3.2. Events as projections. 3.3. Hilbert event-state systems. 3.4. From abstract orthoposets of events to Hilbert lattices. 4. The emergence of fuzzy events in Hilbert space quantum theory. 4.1. The notion of effect. 4.2. Effect-Brouwer Zadeh posets. 4.3. MacNeille completions. 4.4. Unsharp preclusivity spaces. 5. Effect algebras and quantum MV algebras. 5.1. Effect algebras and Brouwer Zadeh effect algebras. 5.2. The Lukasiewwicz operations. 5.3. MV algebras and QMV algebras. 5.4. Quasi-linear QMV algebras and effect algebras. 6. Abstract axiomatic foundations of unsharp quantum theory. 6.1. A minimal axiomatization of unsharp QT. 6.2. The algebraic structure of abstract effects. 6.3. The sharply dominating principle. 6.4. Abstract unsharp preclusivity spaces. 6.5. Sharp and unsharp abstract quantum theory. 7. To what extent is quantum ambiguity ambiguous? 7.1. Algebraic notions of 'sharp'. 7.2. Probabilistic definitions of 'sharpness'. II: Quantum Logics as Logic. Introduction. 8. Sharp quantum logics. 8.1. Algebraic and Kripkean semantics for sharp quantum logics. 8.2. Algebraic and Kripkean realizations of Hilbert event-state systems. 8.3. The implication problem in quantum logic. 8.4. Five polynomial conditions. 8.5. The quantum logical conditional as a counterfactual conditional. 8.6. Implication-connectives. 9. Metalogical properties and anomalies of quantum logic. 9.1. The failure of the Lindenbaum properety. 9.2. A modal interpretation of sharp quantum logics. 10. An axiomatization of OL and OQL. 10.1. The calculi of OL and OQL. 10.2. The soundness and completeness theorems. 11. The metalogical intractability of orthomodularity. 11.1. Orthomodularity is not elementary. 11.2. The embeddability problem. 11.3. Hilbert quantum logic and the orthomodular law. 12. First-order quantum logics and quantum set theories. 12.1. First-order semantics. 12.2. Quantum set theories. 13. Partial classical logic, the Lindenbaum property and the hidden variable problem. 13.1. Partial classical logic. 13.2. Partial classical logic and the Lindenbaum property. 13.3. States on partial Boolean algebras. 13.4. The Lindenbaum property and the hidden variable problem. 14. Unsharp quantum logics. 14.1.
Abstract The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig’s formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper’s first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the effect algebra with respect to the relation of coexistence. In particular, we will see that they can differ quite a lot from usual standard automorphisms, which appear for instance in Ludwig’s theorem. As a byproduct of our methods we also strengthen a theorem of Molnár.
Abstract We propose a special relativistic framework for quantum mechanics. It is based on introducing a Hilbert space for events. Events are taken as primitive notions (as customary in relativity), whereas quantum systems (e.g. fields and particles) are emergent in the form of joint probability amplitudes for position and time of events. Textbook relativistic quantum mechanics and quantum field theory can be recovered by dividing the event Hilbert spaces into space and time (a foliation) and then conditioning the event states onto the time part. Our theory satisfies the full Lorentz symmetry as a ‘geometric’ unitary transformation, and possesses relativistic observables for space (location of an event) and time (position in time of an event).